Bravais Lattice in Two Dimensions

shayaan_musta

In the book Applied Physics by P.K.Mittal, on page#25 under the heading of "Bravais lattice in two dimensions", a paragraph says,
"The number of point groups in two dimensions is 10."
My 1_st question is,
Then how many in three dimensions?

Paragraph further says,
"10 groups in two dimensions produce only 5 Bravais lattice."
What is meant by above statement?

Foder

I believe there are 32 point groups in 3D.

shayaan_musta

Thanks Foder.
Yes I agree with you. But in 2D there are 10 point groups then how will you justify this statement in a simple and easy language so that I could be able to teach other for their difficulties.

Foder

3D have more point groups than 2D because of the extra degree of freedom. I mean, point groups represent symmetry operations, such as inversions, reflections and rotations. When you add an extra dimension you can perform those in new directions not parallel to the original 2D plane, allowing also more combinations of them.

shayaan_musta

Yeah Foder I agree with you.
But I think you are not understanding my question.

Look, there are 10 point groups in 2D which are given below,
C1, C2, C3, C4, C6, D1, D2, D3, D4, D6

Agree?

Then I want to know that how these group can be created?

Means there was any genius who discovered that there are 10 point groups in 2D. So how he did it?
If you give one or two example then may be I would be able to understand that how there are 32 point groups in 3D.

I hope now you can understand my question.
Thanks for hard working for me. GOD bless you.